Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for almost all primes $p$?
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asked Sep 25, 2015 at 0:23
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6$\begingroup$ No. The primes that split in the field $\mathbb{Q}(Q)$, where $2Q=P$ are counterexamples. One expects a positive density of such primes, but not density one, if $P$ is not divisible over the rationals. $\endgroup$– Felipe VolochCommented Sep 25, 2015 at 0:36
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6$\begingroup$ Already in $\mathbb{G}_m$ the answer is clearly negative, for the same reason that Felipe gave you. The statement on positive density can be proved conditionally on GRH. One can in fact say a lot more in the elliptic case: the group $E(\mathbb{F}_p)$ is the cyclic group $\langle P \mod{p} \rangle$ for a definite positive density of primes $p$, unless there is an obvious global obstruction ($P$ being divisible over $\mathbb{Q}$ or the rational torsion being non-cyclic). Those are elliptic variants of Artin's conjecture; Alina Cojocaru has obtained various interesting results in this direction. $\endgroup$– Vesselin DimitrovCommented Sep 25, 2015 at 0:52
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$\begingroup$ Thanks for good comments. I'll write Felipe Voloch's as an answer. $\endgroup$– David LampertCommented Sep 25, 2015 at 13:43
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2 Answers
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No. There is a positive density of primes that split in ℚ$(Q, E[2])$ (where $2Q=P$) and excluding the finitely many primes for which reduction of $E[2]$ isn't injective. For such primes any maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ has even order so reduction of $P$ can't be a generator (since reduction of $P$ is 2-divisible).
answered Sep 25, 2015 at 13:57
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However, it is true that $$f_p=\text{order of $\tilde P$ in $\tilde E(\mathbb F_p)$}$$ cannot be "too small, too often." For example, for every $\epsilon>0$, the series $$ \sum_{p~\text{prime}} \frac{\log p}{p\cdot f_p^\epsilon} $$ converges. (More precisely, the series is ${}\le 3\epsilon^{-1}+O(1)$ as $\epsilon\to0$.)
answered Nov 13, 2015 at 16:37
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$\begingroup$ Thanks, would you please give a reference and/or a few general words to justify this? P.S. I enjoy your books on elliptic curves, thanks for them! $\endgroup$ Commented Nov 13, 2015 at 17:04
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1$\begingroup$ That particular result is a special case of one of the main results of: Murty, Rosen, Silverman, Variations on a theme of Romanoff, Internat. J. Math. 7 (1996), 373-391. Thank you for the kind words about my books. There are also many papers by Murty and others on the elliptic analogue of Artin's conjecture. If you google "elliptic artin conjecture", you'll find some references that will lead you to the literature on the subject. $\endgroup$ Commented Nov 13, 2015 at 17:36
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1$\begingroup$ @DavidLampert: I believe it is also possible to prove that $f_P > p^{1/3}$ for almost all $p$, though I don't know if this has been written up anywhere. Up to an $\epsilon$ in the exponent, this appears in C. Matthews, Counting points modulo $p$ in some finitely generated subgroups of algebraic groups;this is a bit stronger than Theorem 4(b) (with $r = 1$) of the quoted paper. Improving the $1/3$ exponent is an open problem as far as I am aware, though on GRH it is possible to raise it all the way up to $1-\epsilon$. $\endgroup$ Commented Nov 13, 2015 at 18:40